Arithmetic

In this paper, a set of operators suitable for digit-serial FIR filtering is presented. The canonical and inverted forms are studied. In each of these structures both the symmetrical and anti-symmetrical particular cases are also covered. All circuits have been implemented using an EPF10K50 Altera FPGA. Main results show that the canonical form presents the less occupation and the higher throughput. The 8-tap filter versions implemented can be applied in real-time processing with sample rate ranging up to 7MHz using the bit-serial versions and up to 25 MHz with the bit-parallel ones.

Previous research on children's conceptual and procedural understanding of fractions, and other arithmetic skills, has led to contradictory conclusions. Some research suggests that children learn conceptual knowledge before procedural knowledge, some suggests that they learn procedural knowledge before conceptual knowledge, and other research suggests that they learn conceptual knowledge and procedural knowledge in tandem. We propose that these contradictory findings may be explained by considering individual differences in the way that children combine conceptual and procedural knowledge. A total of 318 Grade 4 and 5 students in the United Kingdom (mean age ϭ 9.0 years) completed a measure of fractions understanding, which included subscales of conceptual and procedural knowledge. A cluster analysis identified 5 distinct clusters that differed from each other in terms of their relative success with conceptual and procedural problems. The existence of these clusters suggests that there may be more than one way in which children draw on conceptual and procedural knowledge. Some children rely more on procedural knowledge and others more on conceptual knowledge, but these differences may not be related to developmental processes. The children in the 5 clusters differed in their total fractions performance and in their understanding of intensive quantities. These differences suggest that children who rely on conceptual knowledge may have an advantage compared to those who rely more exclusively on procedural knowledge.

Calculia is considered to be the ability of performing arithmetic operations, the preconditions for the development of mathematical skills in the complex functioning of psychological functions represented in neuro-anatomical systems, as well in the interaction with the environment. Problems in acquiring arithmetic skills can be described as difficulties in counting as well as developmental dyscalculia. Reported prevalence rate for this problem in general population is 6%-10%. The most common difficulties in counting are: difficulties in logic, difficulties in planning, perseverance of inappropriate (responses) procedures and poor understanding of arithmetic operations. This paper aims to identify the influence of assessed arithmetic skills and to clarify to what extent the assessed cognitive skills affect each other in children with difficulties in counting. The relations between the level of working memory and acquiring of arithmetic skills, between the attention and the conceptual...

In this paper, a methodology for the development of fault-tolerant adders based on the radix 2 signed digit (SD) representation is presented. The use of a number representation characterized by a carry propagation confined to neighbor digits implies interesting advantages in terms of error detection, fault localization, and repair. Errors caused by faults belonging to a considered stuck-at fault set can be detected by a parity-based technique. In fact, a carry-free adder preserving the parity of the augends can be implemented allowing fault detection by using a parity checker. Regarding fault localization, the "carry-free" property of the adder ensures the confinement of the error due to a permanent fault to only few digits. The detection of the faulty digit has been obtained by using a recomputation with shifted operands method. Finally, after the fault localization, graceful degradation of the system intended as the reduction of the performances versus a correct output computation can be obtained by using two different procedures. The first one allows obtaining the correct output by recomputing the result performing two different shift operations and using the intersection of the obtained results to recover the correct output, while the second one is based on a reduced dynamic range approach, which allows us to obtain the result in only one step, but with fewer output digits.

To subdivide the axis k into 2 new axes p and q, we place k=p+qN/r with p=0,1, ... , N/r s-1, s=0,1, ... ,logrN-1 and q = 0,1,...., r -1. Therefore, X(k) is replaced using new indices p and q where x [n] is the input sequence, X[k] is the output sequence, N is the transform length, w: k = e-j(2tr/N)nk is called the twiddle factor in the butterfly structure, and j2=_1. Both x [n] and X [k] are complex valued number sequences.

In this paper,w ec onsider the problem of learning to predict the correct pose of a 3D object, assuming orthographic projection and 3D linear transformations. A neural network is trained to learn the desired mapping.F irst, we consider the problem of predicting all possible views that an object can produce.T his is performed by representing the object with a small number of reference views and using algebraic functions of views to construct the space of all possible views that the object can produce.F undamental to this procedurei sam ethodology based on Singular Value Decomposition and Interval Arithmetic for estimating of the ranges of values that the parameterso fa lgebraic functions can assume.Then, a neural network is trained using a number of views (training views) whicha re generated by sampling the space of views of the object. During learning,a training viewispresented to the inputs of the network which is required to respond at its outputs with the parameterso f the algebraic functions used to generate the viewf romt he reference views. Compared to similar approaches in the literature, the proposed approachh as the advantage that it does not requiret he 3D models of the objects or a large number of views, it is extendible to other types of projections, and it is morepractical for object recognition.

Binary image compression is desirable for a wide range of applications, such as digital libraries, map archives, fingerprint databases, facsimile, etc. In this paper, we present a new highly efficient algorithm for lossless binary image compression. The proposed algorithm introduces a new method, direct redundancy elimination, to efficiently exploit the two-dimensional redundancy of an image, as well as a novel

This paper considers the use of punctured quasiarithmetic (QA) codes for the Slepian-Wolf problem. These entropy codes are defined by finite state machines for memoryless and first-order memory sources. Puncturing an entropy coded bit-stream leads to an ambiguity at the decoder side. The decoder makes use of a correlated version of the original message in order to remove this ambiguity. A complete distributed source coding (DSC) scheme based on QA encoding with side information at the decoder is presented, together with iterative structures based on QA codes. The proposed schemes are adapted to memoryless and first-order memory sources. Simulation results reveal that the proposed schemes are efficient in terms of decoding performance for short sequences compared to well-known DSC solutions using channel codes.

The formal analysis described here detects two so far undetected real deadlock situations per thousand C source files or million lines of code in the open source Linux operating system kernel, and three undetected accesses to freed memory, at a few seconds per file. That is notable because the code has been continuously under scrutiny from thousands of developers' pairs of eyes. In distinction to model-checking techniques, which also use symbolic logic, the analysis uses a "3-phase" compositional Hoare-style programming logic combined with abstract interpretation. The result is a customisable post-hoc semantic analysis of C code that is capable of several different analyses at once.

This paper presents new, fast hardware for computing the exponential function, sine, and cosine. The main new idea is to use low-precision arithmetic components to approximate high precision computations, and then to correct very quickly the approximation error periodically so that the effect is to get high precision computation at near low-precision speed. The algorithm used in the paper is a nontrivial modification of the well-known CORDIC algorithm, and might be applicable to the computation of other functions than the ...

Nunes and Bryant (Children doing mathematics, Blackwell, Oxford, 1996) proposed that an understanding of the additive composition of number could be a precursor to an understanding of the decimal structure. If this is so, children should achieve an understanding of additive composition before they can handle the decimal structure. The aim of our study was to determine the developmental timing

In this paper we consider the duality between process and concept in mathematics, in particular using the same symbolism to represent both a process (such as the addition of two numbers 3+2) and the product of that process (the sum 3+2). The ambiguity of notation allows the successful thinker the flexibility in thought to move between the process to carry out a mathematical task and the concept to be mentally manipulated as part of a wider mental schema. We hypothesise that the successful mathematical thinker uses a mental structure which is an amalgam of process and concept which we call a procept. We give empirical evidence from simple arithmetic to show that this leads to a qualitatively different kind of mathematical thought between the more able and the less able, in which the less able are actually doing a more difficult form of mathematics, causing a divergence in performance between success and failure.

Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.

As recent research efforts have focused on preventing reading difficulties and enhancing the effectiveness of special education services for students with reading problems, similar efforts in mathematics have not been realized. This article describes the development and preliminary field testing of a set of measures designed to screen students in kindergarten and first grade to identify those at risk for potential mathematics difficulties. Evidence-based steps to streamline the screening process are described, and plans for testing the predictive validity of the measures are outlined.

Abstract-We describe a sequential universal data compression procedure for binary tree sources that performs the “double mixture.” Using a context tree, this method weights in an ef-ficient recursive way the coding distributions corresponding to all bounded memory tree sources, and ...

The efficacy of an early mathematics program aimed at developing number sense and built entirely on children's invented procedures and on their infotmally acquired quantitative knowledge was studied. To socialize childy,.:n to think of themselves as reasoners about numbers, the classroom program routinely provided daily conversation about numbers and drew attention to quantitative examples in everyday living situations. The program, initiated in 1988-89, was based on the following principles: draw children's J.nformal knowledge into the classroom, develop children's trust in their own knowledge, use formal notations as a public record of discussions and conclusions, introduce the whole additive structure as quickly as possible, tc..1k about mathematics in addition to practicing it, and encourage everyday problem finding. All children in the first through third grade classes in a school serving a largely minority and poor population participated. First graders were interviewed three times during the year. To assess whether the computational aspects of the standard curriculum were being met, data from the standardized mathematics achievement test that the sralool annually gives its first graders at the end of March were examined. The test scores of second and third graders who were introduced to a modified version :f the program part-way through the 1988-89 school year were also assessed. The program produced large improvements both in number sense and in computational competence across all ability levels. Thinking-based programs successfully teach basic number facts and arithmetic procedures that are the corn of the traditional primary mathematics program. An invent.c,n-based mathematics program is suitable even for children who are not socially favored or initially educationally able. (RLC)

This is the ultimate proof that the Prime Numbers are not Random Numbers as famous Mathematicians believe and claim publicly through presentations you can find on U Tube
Any pupil around the world and a wide public will understand the first 2 pages of the paper below.  Feel free to ask questions by emailing me [email protected]

Reading performance has been shown to be linked with balancing skills, possibly indicating an involvement of the cerebellum in reading-related tasks. In our study, we examined whether a similar connection can be detected for arithmetic performance. We assessed basic arithmetic skills of 8-10-year-old children (n ¼ 53) and asked them to balance on the left or right foot, with eyes open or closed. Results revealed substantial correlations between performance in arithmetic tasks and in the balancing tasks with closed eyes even when controlling for attentional and reasoning capabilities. These findings are interpreted in terms of a cerebellar involvement in arithmetic tasks. We propose that verbally mediated arithmetic tasks like multiplication might be related to cerebellar functions in different ways than those arithmetic tasks that require elaboration strategies and quantity manipulations, such as subtraction.

The residue number system (RNS) has been an important research field in computer arithmetic for many decades, mainly because of its carry-free nature, which can provide high-performance computing architectures with superior delay specifications. Recently, research on RNS has found new directions that have resulted in the introduction of efficient algorithms and hardware implementations for RNS with much better performance than previous ones. Furthermore, the applicability of RNS in many computer and digital signal processing (DSP) applications has simultaneously greatly increased. Hence, the need is evident for the development of a new and well-organized RNS teaching method with emphasis on recent achievements. In this paper, a step-by-step teaching process for RNS that describes RNS design in separate parts is presented. Each part is investigated in detail, taking into account the recent research performed on RNS. The compatibility of the proposed method with the new RNS research trends makes this method suitable for researchers as well as students.

This article focuses on the kinds of conceptual changes that occur when students have to deal with negative numbers in elementary algebraic operations. Interviews were carried out with twelve 8th-grade level students who were selected on the basis of their results in a test where they were required to reduce polynomials. The questions applied to their strategies and to the meaning they gave to the minus sign.

A r q. R o b e r t o S a l d i v a r O l a g u e. G r a d u a d o e n I C S , E s c u e l a d e A r q u i t e c t u r a , S c r a n t o n P. a. , U S A. Titulado en Arquitectura en el Instituto Tecnológico de Zacatecas, México M a y o d e l 2 0 1 3

All rights reserved 1 2 3 4 13 12 11 10 This volume is a product of the staff of the International Bank for Reconstruction and Development / The World Bank. The findings, interpretations, and conclusions expressed in this volume do not necessarily reflect the views of the Executive Directors of The World Bank or the governments they represent. The World Bank does not guarantee the accuracy of the data included in this work. The boundaries, colors, denominations, and other information shown on any map in this work do not imply any judgement on the part of The World Bank concerning the legal status of any territory or the endorsement or acceptance of such boundaries.

An efficient method for computing the discrete cosine transform (DCT) is proposed. Based on direct decomposition of the DCT, the recursive properties of the DCT for even length input sequence is derived, which is a generalization of the radix 2 DCT algorithm. Based on the recursive property, a new DCT algorithm for even length sequence is obtained. The proposed algorithm is very structural and requires fewer computations when compared with others. The regular structure of the proposed algorithm is suitable for fast parallel algorithm and VLSI implementation.

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students’ construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in relation to the whole. Students’ whole number multiplicative concepts were found to be critical constructive resources for students’ fraction composition schemes. Specifically, the interiorization of two levels of units, a particular multiplicative concept, was found to be necessary for the construction of a unit fraction composition scheme, while the interiorization of three levels of units was necessary for the construction of a general fraction composition scheme. These findings contribute to previous research on students’ construction of fraction multiplication that has emphasized partitioning and conceptualizing quantitative units. Implications of the findings for teaching are considered.

The difficulties associated with translating a set of scores into letter-grades are discussed. A novel method for automating this process, the Fuzzy Grading System, is developed and compared to traditional practices. Letter-grades are recognized to be f u u y descriptors of students' performance. Thus, operations aimed at defining letter-grade bouhdaries are naturally carried out in the context of fuzzy set theory and logic. The Fuzzy Grading System utilizes students' and instructor's performance measures in order to m d i y a set of collectively approved, a priori fuzzy grades, so as to produce a "fair" mark distribution. The validity of grades is increased by compensating for factors that are not directly accessible to simpler, traditional grading methods.

In this paper, we present new algorithms for the calculation of convolutions by means of number theoretic transforms over modulo rings. Not only are these algorithms more efficient than currently used methods, but they are also very flexible. Indeed, using special algorithms for short convolutions allows trading computational efficiency for structural simplicity.

Recent studies have shown that the use of educational games during learning process is dramatically increased. Furthermore, researchers suggest the attachment of adaptive features in order to motivate students and assess their knowledge level on a specific educational subject. In this paper, we present an educational browser-based game with coins that contributes to understanding better the addition process in elementary education. The game encompasses user modeling and adaptive techniques. It determines students’ knowledge level and helps them outcome difficulties and obtain fluency in arithmetic addition skills.

The efficacy of an early mathematics program aimed at developing number sense and built entirely on children's invented procedures and on their infotmally acquired quantitative knowledge was studied. To socialize childy,.:n to think of themselves as reasoners about numbers, the classroom program routinely provided daily conversation about numbers and drew attention to quantitative examples in everyday living situations. The program, initiated in 1988-89, was based on the following principles: draw children's J.nformal knowledge into the classroom, develop children's trust in their own knowledge, use formal notations as a public record of discussions and conclusions, introduce the whole additive structure as quickly as possible, tc..1k about mathematics in addition to practicing it, and encourage everyday problem finding. All children in the first through third grade classes in a school serving a largely minority and poor population participated. First graders were interviewed three times during the year. To assess whether the computational aspects of the standard curriculum were being met, data from the standardized mathematics achievement test that the sralool annually gives its first graders at the end of March were examined. The test scores of second and third graders who were introduced to a modified version :f the program part-way through the 1988-89 school year were also assessed. The program produced large improvements both in number sense and in computational competence across all ability levels. Thinking-based programs successfully teach basic number facts and arithmetic procedures that are the corn of the traditional primary mathematics program. An invent.c,n-based mathematics program is suitable even for children who are not socially favored or initially educationally able. (RLC)

In a 3-year longitudinal study, 471 children were classified, based on their performances on arithmetic tests in first and second grade, as having persistent arithmetic disabilities (AD), persistent low achieving (LA), persistent typical achieving, inconsistent arithmetic disabilities (DF1), or inconsistent low achieving in arithmetic. Significant differences in the performances on the magnitude comparison in kindergarten (at age 5-6) were found between the AD and LA and between the AD and DF1 groups. Furthermore, the percentage of true-positive AD children (at age 7-8) correctly diagnosed in kindergarten by combination of procedural counting, conceptual counting, and magnitude comparison tasks was 87.50%. When composing clinical samples, researchers should pay attention when stipulating restrictive or lenient cutoffs for arithmetic disabilities and select children based on their scores in 2 consecutive years, because the results of studies on persistent low achievers or children with inconsistent disabilities cannot be generalized to children with persistent arithmetic disabilities.

""Sharing a field into plots of equal area is one of the oldest known mathematical problems. Some of these sharing problems lead to wonderful arithmetic problems that seem to have excited the curiosity of Mesopotamian mathematicians. This article aims to show the link between sharing fields and arithmetic riddles through some samples of mathematical cuneiform tablets dating mainly from Old Babylonian period (early second millennium BC).

Partager un champ en plusieurs parcelles d’aires égales est un des plus anciens problèmes mathématiques connus en Mésopotamie. Certains partages particuliers conduisent à de très jolis problèmes d’arithmétique qui semblent avoir excité la curiosité des mathématiciens mésopotamiens. Cet article se propose de montrer le lien entre des problèmes d’arpenteur et des problèmes d’arithmétique au travers de quelques exemples de tablettes mathématiques d’époque paléo-babylonienne (début du deuxième millénaire avant notre ère).""

"Elementary Arithmetic in Base 7" presents a novel framework that integrates numerical systems with color representation through a unique chromatic approach based on cyclic orders. The concept revolves around the relationship between different numerical bases and their visual representations, specifically focusing on Base 7. The paper explores the mathematical properties of cyclic numbers, monotonic functions, and their implications for establishing a harmonic system that maps numbers to colors in a structured way, thereby creating a cohesive language for sounds, colors, and numbers. For more information see the following link: https://pdpseven.wixsite.com/sound-color.
For more information see the following link:
https://pdpseven.wixsite.com/sound-color

Children's symbolic number sense was examined at the beginning of first grade with a short screen of competencies related to counting, number knowledge, and arithmetic operations. Conventional mathematics achievement was then assessed at the end of both first and third grades. Controlling for age and cognitive abilities (i.e., language, spatial, and memory), number sense made a unique and meaningful contribution to the variance in mathematics achievement at both first and third grades. Furthermore, the strength of the predictions did not weaken over time. Number sense was most strongly related to the ability to solve applied mathematics problems presented in various contexts. The number sense screen taps important intermediate skills that should be considered in the development of early mathematics assessments and interventions.

We examined elementary school teachers' justifications to number-theoretical "for all" propositions and existence propositions, some of which are true while others are false. We also assessed whether teachers regarded their justifications as mathematical proofs. About half of the teachers produced formal algebraic proofs. A smaller number of teachers produced non-formal proofs appropriate for presentation in elementary school classes. However, a substantial number of teachers applied inadequate methods to validate or refute the propositions. Finally, many teachers were uncertain about the status of the justification they gave. (Author)

Fast bit-reversal algorithms have been of strong interest for many decades, especially after Cooley and Tukey introduced their FFT implementation in 1965. Many recent algorithms, including FFTW try to avoid the bit-reversal all together by doing in-place algorithms within their FFTs. We therefore motivate our work by showing that for FFTs of up to 65.536 points, a minimally tuned Cooley-Tukey FFT in C using our bit-reversal algorithm performs comparable or better than the default FFTW algorithm.

This paper describes a limited longitudinal study of young children's early numeracy development within three testing cycles, at the mid-point and towards the end of their reception year (at five years-of-age) and again at the mid-point of Year 1 (at six years-of-age), located within the broader context of progress through to Key Stage 1 SAT results (at seven years). Assessment was carried out using the Utrecht Early Mathematical Competence Test (Van Luit et al., 1994). This comprised eight sub-topics, five items in each, including comparison, classification, correspondence, seriation, counting, calculation and real-life number problem solving. Broadly, one set of sub-tests related to understanding of relations in shape, size, quantity and order, whilst a second set of sub-tests related to basic arithmetic. Three hundred pupils were selected from twenty-one schools, large and small, from rural and urban areas, with high and low concentrations of children eligible for free school meals and/or with special educational needs, as well as representing a broad range of achievements levels based on standards assessment tasks. Whilst this paper focuses upon the performance of English pupils, reference is also made to the larger European sample which involved children from Flemish-speaking Belgium, Germany, Greece, Slovenia and the Netherlands. Results showed that children's total scores at around the mid-point of reception year were indeed predictive of later achievement at the end of KS1 though the combined scores over three testing cycles which extended to the mid-point of Year 1, were more so. Discriminant analysis confirmed that a combination of a counting sub-test (one seemed sufficient) and a sub-test focusing on understanding of relations in shape, size, order or quantity (a different one at each testing cycle), together with the general number knowledge sub-test was best predictive of final SAT levels. Comparison with the international data set suggested a different trajectory for English pupils, with more of a bias towards arithmetic sub-tests than their European counterparts who start school later. Moreover, the pattern of dependence of scores on age in

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